Question 848·Medium·Math

0 / 2000

A laboratory technician recorded the mass of a bacterial culture at 1-hour intervals. The results are shown below.

Time, $t$ (hours)	0	1	2	3	4
Mass, $m$ (grams)	2.0	2.9	4.3	6.0	8.8

A scatterplot of the data (not shown) suggests an exponential growth pattern. Which equation best models the mass $m$ in grams as a function of time $t$ ?

An exponential model fits data that multiply by about the same factor over equal time intervals. For measured data, the ratios may not match exactly, so an exponential regression in Desmos uses all the readings to estimate the starting value and hourly factor. Use a repeated multiplier, not a fixed amount added each hour.

Hints

Hint 1
In the exponential model $m=ab^t$ , $a$ is the mass at hour $0$ , and $b$ multiplies the mass each time the exponent rises by $1$ . Since readings are one hour apart, what does $t$ count?
Hint 2
A regression fits a model to all the measurements rather than treating one rounded ratio as exact. Put time in $x_1$ and mass in $y_1$ . Which answer has a starting value and multiplier near the fitted values?

Step-by-step

Approach 1: Fit the measurements in Desmos

Step 1
Write the kind of model to fit
The prompt calls for exponential growth, which means the mass is multiplied by a factor for each equal time step. The readings are one hour apart, so $t$ counts those steps. Write the model as $m=ab^t$ , where $a$ is the starting mass and $b$ is the multiplier per hour.
Step 2
Fit all five measurements
Enter the hours in Desmos's $x_1$ column and the matching masses in $y_1$ . Then type $y_1\sim ab^{x_1}$ ; the tilde tells Desmos to fit the model to the table. Under PARAMETERS, Desmos reports approximately $a=2.01$ and $b=1.45$ . These are estimates because the measurements don't follow an exact ratio.
Step 3
Match the fitted numbers to a model
The fitted $a$ is near $2$ , and the fitted hourly multiplier $b$ is near $1.5$ . Choose those numbers in $m=ab^t$ . The model needn't hit every measured mass exactly: $m=2(1.5)^t$ is the best choice for the mass in grams after $t$ hours. Choice C.

Approach 2: Estimate the multiplier from the table

Step 1
Use the starting mass
At $t=0$ , the table gives $2.0$ grams. Since $b^0=1$ , the number in front of $b^t$ must be the initial value, $2$ grams. So the model has the form $m=2b^t$ .
Step 2
Compare masses one hour apart
For an hourly multiplier, divide a mass by the mass one hour earlier. Use an early pair and a late pair to see whether the estimate holds up:
$\frac{2.9}{2.0}=1.45$ $\frac{8.8}{6.0}\approx1.47$
Both are near $1.5$ . The small difference is why you shouldn't treat one measured ratio as exact. The best model is $m=2(1.5)^t$ grams. Choice C.

Question 848·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this